Past exam of the short options course of the University of Oxford / 2023 / 8 (Special relativity)

Use the Minkowski metric $\eta =\mathrm{diag}(1,-1,-1,-1)$ and coordinates $x^0=ct$. At a point, a smooth path is timelike if its tangent $\dot{x}$ is a timelike vector, here meaning ${\eta }_{ab}{\dot{x}}^a{\dot{x}}^b>0$. It is lightlike if its tangent is a null vector: the same expression is zero and the tangent is nonzero. In these coordinates, a future-directed timelike tangent has spatial speed less than $c$; a lightlike tangent has speed $c$.

For increasing parameter $s$ along a future-directed timelike curve, the elapsed proper time isor $\int \sqrt{1-|\mathbf{v}{|}^2/c^2}dt$ when coordinate time is the parameter. A Lorentz transformation satisfies $L^T\eta L=\eta$, so replacing $\dot{x}$ by $L\dot{\tilde{x}}$ leaves the expression under the square root unchanged. Hence the integral is invariant. A monotone change of path parameter also leaves it unchanged by the ordinary substitution rule.

For the collinear boosts specified in the question, multiplication of the two $2$-by-$2$ blocks and the hyperbolic addition formulas giveThus the resulting rapidity is $\boxed{{\theta }^{\prime \prime }=\theta +{\theta }^{\prime }}$. Collinearity matters: boosts in different spatial directions do not in general combine into a pure boost of this form.

Using $v/c=\tanh \theta$ and the hyperbolic tangent addition identity,Consequently the relativistic velocity addition rule is

When $0\le v,v^{\prime }\ll c$, expand the denominator:The leading term is ordinary Galilean velocity addition. When both $v,v^{\prime }\to c$ from below, the exact formula instead gives $\boxed{v^{\prime \prime }\to c}$, rather than $2c$. Indeed, writing $a=v/c$, $b=v^{\prime }/c$, we have $1-v^{\prime \prime }/c=(1-a)(1-b)/(1+ab)>0$ whenever $a,b<1$.

The speed of one rocket in the other's rest frame is obtained by relativistic velocity addition:The Lorentz factor is $\gamma =(1-u^2/c^2{)}^{-1/2}=5/3$. By length contraction, the spatial length measured at simultaneous times in that rocket's rest frame isThis is the inertial-frame length measurement; a visual image assembled from light arriving at one instant is a different measurement.